Some recent work on the curvaton paradigm

Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35 www.elsevierphysics.com

Some recent work on the curvaton paradigm David H. Lytha a

Physics Department Lancaster University Lancaster LA1 4YB UK

I review the simplest curvaton scenario. Using a pseudo Nambu-Goldstone boson as curvaton avoids the ηproblem of inflation which plagues most curvaton candidates. Then I report on a collaboration with E. J. Chun and K. Dimopoulos, describing a concrete realization of the curvaton mechanism. It uses a pseudo Nambu-Goldstone boson can be found in the supersymmetric Peccei-Quinn mechanism resolving the strong CP problem. In the flaton models of Peccei-Quinn symmetry breaking, the angular degree of freedom associated with the QCD axion can naturally be a flat direction during inflation and provides successful curvature perturbations. In this scheme, the preferred values of the axion scale and the Hubble parameter during inflation turn out to be about 1010 GeV and 1012 GeV, respectively. Moreover, it is found that a significant isocurvature component, (anti)correlated to the overall curvature perturbation, can be generated, which is a smoking-gun for the curvaton scenario. Finally, non-Gaussianity in the perturbation spectrum at potentially observable level is also possible.

1. The curvaton paradigm The primordial curvature perturbation is caused presumably by some scalar field, which acquires its perturbation during inflation. For a long time it was generally agreed that the curvature-generating field would be generated during inflaton [1] (the inflaton paradigm). Then it was suggested instead that the responsible field is some ‘curvaton’ field which ha nothing to do with inflation [2,3] (see also [4]). The curvaton generates the curvature perturbation only when its density becomes a significant fraction of the total. The curvaton proposal has received enormous attention because it opens up new possibilities both for model-building and for observation.1 According to the inflaton paradigm, the curvature perturbation is constant from the end of inflation onwards. In that case the primordial curvature perturbation is given by ζ(k, t) =

H δφ φ˙

(1)

1 More

recently still it has been suggested that the curvaton acts by causing inhomogeneous reheating [5] or through a preheating mechanism [6].

0920-5632/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.nuclphysbps.2005.04.047

with δφ the perturbation in the field pointing along the inflaton trajectory at the end of inflation. If the inflaton paradigm is abandoned, the curvature perturbation can grow after the end of inflation. It is then natural to go to the other extreme and suppose that the curvature perturbation at the end of inflation is negligible. The field(s) responsible for the curvature perturbation can be any field different from φ at the end of inflation. It might be one of the fields in the inflation model, but usually it is supposed to be a field which has no significant effect on the dynamics of inflation. Whatever its origin, the curvature perturbation is given as a function of time by

ζ(k, t) =




N,n δφn (k),

(2)

n

in terms of the perturbations in one or more light fields evaluated a few Hubble times after horizon

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D.H. Lyth / Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35

exit. Here,2 N,n ≡

∂N (φ1 , φ2 , · · · , k, ρ) , ∂φn

(3)

where N is the number of e-folds of expansion from the epoch of horizon exit k = aH to the epoch at which ζ is to be evaluated, determined by the value of the energy density ρ. This leads to the spectrum 2
 Hk |N,n |2 (4) Pζ (k, t) = 2π n

where Hk is the Hubble parameter at horizon exit, and to the spectral index n(k, t)−1 = −2−

2

+2 MP2 N,n N,n

ηnm N,n N,m ,(5) N,j N,j

where identical indices are summed over. Here, 2 ˙  ≡ −H/H and ηnm ≡

∂2V . ∂φn ∂φm

(6)

What we are doing here is to distinguish between the case that the curvature perturbation levels out to a constant value at the end of inflation, and the case that it is negligible during inflation rising to a constant value only later. The intermediate case is of course possible, but we shall not consider it. We are assuming that the curvature perturbation is generated by a single field which we denote by σ. Its spectral index is n − 1 = 2ησσ − 2 ,

(7)

ησσ ≡

,

(8)

with m2σ ≡ (∂ 2 V /∂σ 2 ) the effective mass-squared of σ. The contribution of  to the spectral index is negligible in typical inflation models. Also, ησσ is 2 To

ζσ ≡

1 δρσ = const , 3 ρσ

(9)

where the curvaton energy density ρσ is averaged over an oscillation and its perturbation is evaluated on flat slices. Since ρσ is proportional to σ 2 and ζσ is constant during the oscillation we may write

δσ δρσ 2 , σ ρσ

where m2σ 3Hk2

expected to be negligible making n indistinguishable from 1. This is because σ has nothing to do with the inflation dynamics and does not ‘know’ about the end of inflation. At the moment, two alternatives to the inflaton paradigm have been proposed. In one of them, the perturbation in some ‘curvaton’ field generates a perturbation in the curvaton energy density and this is the source of the curvature perturbation. In the other, some ‘modulon’ field alters the mass and/or decay rate of the particle responsible for reheating, generating thereby the curvature perturbation even though its energy density never becomes significant. We shall describe only the curvaton paradigm. The crucial assumption is that the curvaton oscillation takes place in a background of radiation. The curvaton energy density is supposed to be negligible until after the oscillation begins, and with it the curvature perturbation. These quantities grow during the oscillation because ρσ /ρr is proportional to a(t). To follow the growth of the curvature perturbation we can consider the separate matter and radiation contributions as in [29], with ζr = 0 and ζm = ζσ . The latter is constant

be more precise, Eq. (2) is valid if the curvature perturbation depends linearly on the field. One needs to go beyond the linear approximation to deal with nongaussianity on cosmological scales, and to deal with the large small-scale curvature perturbation present in some inflation models.

(10)

where the right-hand side is evaluated at the beginning of the The curvaton decays before cosmological scales start to enter the horizon, and just before decay the curvature perturbation is ζ≡

1 δρσ = rζσ 3ρ+P

zetaof r

,

(11)

with r ≡ ρσ /(ρ + P ) ≤ 1. If the curvaton potential is sufficiently flat, σ has negligible evolution before the oscillation starts. Then the spectrum

D.H. Lyth / Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35

of the curvature perturbation just before decay is given by 1/2

Pζ

= 2r

H∗ 2πσ∗

curvspec

.

(12)

where the star means that quantities are to be evaluated during inflation. The decay products of the curvaton are supposed to thermalize completely with the preexisting radiation (unless the latter is completely negligible, corresponding to r very close to 1), to produce thermal equilibrium where everything is determined. After the curvaton decay, there is supposed to be essentially complete thermal equilibrium so that everything is determined by the temperature, leading to the conservation of the curvature perturbation.3 The prediction Eq. (12) depends on the mean value σ∗ within our Universe. If we occupy a typical position in the Universe, and if inflation has lasted long enough when we exit the horizon, the probability for a given value will be given by   8π 2 V (φ) f prob ,(13) F = A exp − 3H 4

where A is a normalization constant making ∞ F dφ = 1. 0 Assuming that our location is typical this then roughly determines σ∗ , giving a more or less definite prediction for the spectrum curvature perturbation. Because the properties required of it are undemanding, the curvaton field can rather easily be identified with a field already proposed for some other reason, within an extension of the Standard Model. As a result one can envisage the possibility that a future collider experiment might shed light on the origin of the primordial density perturbation. Such a possibility hardly exists within the inflaton paradigm. 2. The Peccei-Quinn proposal In all cases, the field responsible for the curvature perturbation must be light during inflation, 3 Small

departures from this state of affairs are allowed, producing for instance a primordial isocurvature perturbation, as long as the curvature perturbation is conserved.

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in the sense that its effective mass meff is much less than the Hubble parameter H. (To be pre2 cise, we need during inflation m2eff < ∼ 0.1H , so 2 2 2 that the spectral tilt 1 − n  3 meff /H satisfies the observational constraint.) If the responsible field is a curvaton, it should preferably also remain light after inflation, until H falls below the true mass [7]. These requirements are in mild conflict with the generic expectation from supergravity, that the mass-squared of each scalar field will be at least of order H 2 [8]. This is similar to the famous η-problem of inflation. Ways have been proposed to keep the responsible field sufficiently light [1], the most straightforward of them being to make it a pseudo Nambu-Goldstone boson (PNGB) (for the curvaton see [2,9]). For economy, and also to facilitate contact with observation, one would like a candidate for the responsible field to be one that is present in an already-existing model, designed for some purpose other than the generation of the curvature perturbation. Several such candidates have been proposed [9–13], but in general they do not come with a satisfactory mechanism for keeping the curvaton sufficiently light. The purpose of [14] is to suggest a natural curvaton candidate, which is present in flaton models of Peccei-Quinn symmetry breaking, and which is a PNGB. The Peccei-Quinn (PQ) symmetry provides a nice solution to the strong CP problem [15]. It is an anomalous global symmetry, U (1)P Q , spontaneously broken at an intermediate scale fP Q with preferred value around 1012 GeV providing enough dark matter (axion) of the universe. Its PNGB is the axion, which must be extremely light to satisfy the observational constraints [16]. In the context of supersymmetry (SUSY), two complex fields (or more) are needed to implement a global symmetry due to holomorphicity of the superpotential (unless the symmetry in question is R–symmetry). Hence, in SUSY, to spontaneously break the PQ symmetry one needs at least two complex fields, corresponding to two radial and two real angular degrees of freedom. One of the latter is the axion, and the other is our curvaton candidate.4 We can define a symmetry 4 By

‘degree of freedom’ we mean as usual a normal mode

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D.H. Lyth / Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35

(explicitly broken of course) acting only on the combination of phases which corresponds to the curvaton, and then the curvaton is the PNGB of that symmetry. To understand how our model works, recall first that two fundamentally different paradigms exist for the PQ symmetry breaking fields. In one of them, the potentials involve only renormalizable terms in the superpotential plus small soft SUSY breaking terms, leading to a normal Mexican-hat potential whose height and width have the same scale fP Q . According to this paradigm, PQ symmetry breaking persists in the limit of unbroken SUSY so that the axion has a welldefined supersymmetric scalar partner called the saxion (as well as a fermionic partner called the axino). The saxion is the only light degree of freedom apart from the axion, its mass coming from SUSY breaking and being of order TeV for gravity-mediated SUSY breaking. The other two degrees of freedom are heavy with mass of order fP Q . According to the other paradigm, the PQ symmetry breaking fields are instead flaton fields [17], so-called because they are symmetry-breaking fields, which correspond to flat directions of the potential (i.e., directions in which the quartic term is negligible). Their potential contains only soft SUSY breaking terms and nonrenormalizable terms, which means that it is very flat. In the limit of unbroken SUSY there would be no spontaneous breaking of the PQ symmetry. A nice feature of this paradigm is that the intermediate axion scale fP Q is not put by hand but is generated through a geometric mean of the SUSY breaking scale and the Planck scale. The other degrees of freedom accompanying the axion get their mass only from SUSY breaking, making them of order TeV for gravity-mediated SUSY breaking. In this paper, we point out that this kind of model contains a natural curvaton candidate as the angular degree of freedom. The radial degrees of freedom are less suitable for this purpose, because in the early Universe they presumably acquire the mass of order H which is of the coupled oscillations of the fields, which after quantization corresponds to a particle species.

generic [8] for scalar fields in a supergravity theory. In contrast, a mass of order H is unlikely to be generated for the angular degree of freedom, because it would correspond to a generalized Aterm which is forbidden if the fields responsible for the energy density are charged under certain symmetries [8]. We will analyze how the model parameters like the axion scale, the Hubble parameter and the curvaton decay temperature are constrained to produce successful curvature perturbations. 3. Supersymmetric realization of the PQ and curvaton mechanisms The minimal model of the type that we are considering is a variation of the non-renormalizable superpotential [18–20], which introduces two fields P and Q charged under PQ symmetry and one singlet S allowing the following superpotential S n+3 P QS n+1 P n+1 (14) + λ H H + λ W =h 2 1 2 1 MPn MPn MPn

where the U (1)P Q and ZN charges are assigned respectively as follows; H1 1 2 (n

+ 1) 1

H2 1 2 (n

+ 1) 1

P

Q

S

−1 +1 1 α2

0 α

(15)

with α ≡ e2πi/N and N = n + 3. The field S is supposed to have a negative soft mass-squared of order the gravitino mass m3/2 . This forces all the fields P , Q and S to develop the vacuum expectation values of the order 1/n+1  v ∼ m3/2 MPn . (16) This sets the intermediate PQ scale fP Q ∼ 3 × 1010 , 1 × 1013 , 3 × 1014 GeV

(17)

for n = 1, 2, 3, respectively, and m3/2 = 300 GeV. The negative mass-squared of the S field can come from the renormalization effect [18] or from the initial condition of soft masses. We do not assume the negative mass-squared for the P and Q fields, which would drive the scalar potential unbounded from below when some of the fields P , Q or S are set to zero.

D.H. Lyth / Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35

Note that the superpotential of Eq. (14) provides a natural solution to the µ problem [21] as the first non-renormalzable term leads to the right order of magnitude for the Higgs mass parameter; µ=h

vPn+1 ∼ m3/2 . MPn

The above interaction will be the main source of the decays of the particles corresponding to P , Q or S into the standard (s)particles. The model under consideration contains six gauge singlet scalar particles. Three radial fields get masses of order m3/2 from soft supersymmetry breaking. Among three phase fields, ϕP , ϕQ and ϕS , one combination, ϕP +ϕQ −2ϕS , gets the mass of order m3/2 as the radial fields. The combination, ϕP − ϕQ , is the axion. The other combination is our curvaton candidate, ϕP +ϕQ +ϕS , which becomes massive due to the soft supersymmetry breaking A term; Vsof t = Aλ1

S n+3 P QS n+1 + h.c. , + Aλ2 n MPn MP

inducing also in the vacuum a curvaton mass of order m3/2 . (For a detailed calculation in a similar model of the mass spectrum, see [22].) Note that, in our scheme, the curvaton parameters like its amplitude and decay temperature are fixed by the PQ scale and the µ term, as will be shown explicitly. 4. The potential in the early Universe In order to consider the basic features of the curvaton field σ associated with the QCD axion, let us simplify the original superpotential term in Eq. (14) as follows; λ φn+3 , W (φ) = n + 3 MPn

(18)

where the complex field φ may be thought of as containing the phase field σ and one radial component. This would be literally correct if the vacuum expectation values (VEVs) of P , Q and S happened to be equal, allowing the parameterization, √ (19) φ ≡ |φ|eiθ = |φ| exp(iσ/ 2v)

29

√ with v = φ and σ ≡ 2 vθ. (As noted in the Conclusion, our curvaton model can also be implemented using only φ, though one then loses the connection with the QCD axion.) With the above superpotential (18), the general scalar potential in the early universe can be written as

V

= (3Cφ H 2 − m2φ )|φ|2   λ φn+3 + h.c. + (CA H + A) n + 3 MPn

+ λ2

|φ|2n+4 , MP2n

(20)

where mφ and A are soft supersymmetry breaking masses at zero temperature. Note that we have put negative mass-squared for the φ field at zero temperature to generate the PQ scale as discussed previously. The terms with H are expected to arise from the supersymmetry breaking effect during and after inflation [8], except during radiation domination [23]. The generic expectation for Cφ is |Cφ | ∼ 1, and we are going to assume Cφ ∼ −1 so that there is PQ symmetry breaking. As is well known, a value |CA |  1 can be achieved provided that relevant field values are small on the Planck scale, if a symmetry forbids low-order terms in the K¨ahler potential which are linear in the field responsible for the energy density [8]. To be safe though, we need an estimate of the actual value of CA , considering for completeness also Cφ . During inflation, the supergravity potential is taken to be generated by a single F –term FI ;   2 |W |2 K/MP 2 = 3H 2 MP2 , (21) |FI | − 3 2 V =e MP

where |FI |2

 ∗ ∂I W + W ∂I K/MP2 K II ∗  × ∂I W + W ∂I K/MP2 

≡

∗

(22)

and K II is an element of the inverse of the K¨ahler metric KIJ∗ ≡ ∂I ∂J∗ K with J running over all of the scalar fields. Depending on the model, the field I might be the inflaton, the waterfall field or some other field [1]. We make the following assumptions; (i) in the K¨ahler potential, terms linear in I are negligible

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D.H. Lyth / Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35

(this can be a consequence of a global or gauge symmetry), (ii) the field I is much smaller than the Planck mass; I  MP , as is true in most inflation models, and (iii) the field φ is also much smaller than MP ; φ  MP . The last two conditions allow us to make a perturbative expansion of the potential in terms of I and φ. Under these assumptions, we write the most general K¨ahler potential and superpotential : c K = I † I + φ† φ + 2 I † Iφ† φ MP   d † I IW (φ) + h.c. + MP3 W = W (I) + W (φ) (23)

neglecting terms with higher powers of I † I, φ† φ and W (φ) which give more suppressed contributions. The conditions (ii) and (iii) imply that √ FI ≈ ∂I W ≈ 3HMP and W/MP ≡ αIH  FI with α < ∼ 1. Then, we have

(1 − c) √ I √ H − 3d (24). 3 1 − c(n + 3) − 3α CA ≈ MP MP As advertised, one obtains |Cφ | ∼ 1 but |CA |  1 [8]. If H replaces mφ , Eq. (16) becomes v ∼ (HMPn )1/n+1 , which leads to the following, respectively during inflation, after inflation and after reheating; ⎧ ⎨ (HMPn )1/n+1 (25) v ∼ [max(H, m3/2 )MPn ]1/n+1 ⎩ (m3/2 MPn )1/n+1 = fP Q . Cφ

≈

While the radial degree of freedom follows such a thermal history, the corresponding phase degree of freedom, σ, will enjoy the following potential of the pseudo Nambu-Goldstone boson (PNGB) type [9]: n   σ 

v 3 .(26) 1 − cos V (σ) ≈ (CA H+A)v v MP

which is very flat during and after inflation. From Eq. (25), one finds m2σ

∼ ∼

(CA H + A) max{H, m3/2 } ⎧ ⎨ CA H 2 max(CA H 2 , AH, Am3/2 ) ⎩ Am3/2

where we assumed H∗ > A/CA .

that,

during

inflation

5. The curvaton cosmology It is now obvious that the field σ will naturally realize the curvaton paradigm. Taking Eq.(18) √ literally, σ can be parameterized as θ ≡ σ/ 2v, that is, φ = veiθ [cf. Eq. (19)]. To obtain simple predictions, let us assume that the value of σ is completely randomized during inflation, so that in our part of the Universe θ at horizon exit has some value θ0 < 2π. (We will come back to this point later). It remains constant as far as H mσ and the oscillation starts at H ∼ m3/2 . Finally, the curvaton decays and thermalizes at H ∼ Γσ which is determined to be quite late due to its axionlike couplings. The interaction of σ with ordinary particles is governed by the effective µ term n in Eq. (14) leading to µ = hfPn+1 Q /MP ∼ m3/2 . Thus, the decay rate of σ into two Higgs field is 3

Γσ ≈

(n + 1)2 m3/2 , fP2 Q 4π

(28)

where we put µ = mσ = m3/2 . Taking the approximation of Γσ ∼ m33/2 /fP2 Q and fP Q ∼ (m3/2 MPn )1/n+1 , we get the decay temperature of σ as follows;  160 GeV for n = 1 (29) Tdec ∼ 0.6 GeV for n = 2 30 MeV for n = 3 with m3/2 = 300 GeV. Here we have restricted ourselves to n ≤ 3 for which the decay occurs well before the nucleosynthesis occurring around TBBN ∼ 1 MeV. The density perturbation driven by the curvaton is given by [2] ζ=

1 r H∗ π 4 + 3r σ∗

(30)

where r is the density ratio of ρσ /ργ at the time of the σ decay: (27)

1 r≈ 6



Hm ¯ Γσ

1/2 

σ∗ MP

2 .

(31)

D.H. Lyth / Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35

Here Hm ¯ ≡ min(mσ , ΓI ) [7] where ΓI is the inflaton decay rate determining the reheating temperature. Note that, in our model, the values of r, H∗ and σ∗ are determined by dimensionality of the superpotential in Eq. (18), which are related to the axion scale, fP Q . Inserting Eqs. (25) and (28) to Eqs. (30) and (31), we can determine the inflation Hubble parameter H∗ and the density ratio r which reproduce the observed value of ζ ≈ 5 × 10−5 . First, the value of r can be calculated from the relation:  n2  n4   n+2 MP 2(n+1) 4πζ θ0n+1 Hm r 2 ¯ (32) ∼ m3/2 m3/2 6n/2 1 + 34 r

and then it fixes the inflation Hubble parameter as n+1  3  n n+1 r 1 + H∗ 4 . (33) ∼ (4πζ θ0 ) n r MP

31

primordial tensor perturbation. (Actually, the curvaton model requires a somewhat smaller limit [25], assuming that inflation is of the slow-roll type.) The case with n = 1 then gives most satisfactory result with H∗ ∼ 1012 GeV. In this case, the curvaton decay temperature is Tdec ∼ 100 GeV and the resulting entropy dumping is not significant (it is comparable to the preexisting one), so that the usual cosmological properties are retained. That is, the conventional dark matter candidate, the lightest neutralino or the axion with fP Q = 3 × 1010 GeV, is equally acceptable within our curvaton context, and the usual baryogenesis mechanism can also work. Note, also, that in this case we can have substantial non-Gaussianity in the density perturbation spectrum. Indeed, from Eq. (34) we readily obtain [28]

5 −4/3 . (36)  0.9 θ0 fNL = 4r For a typical set of input parameters; m3/2 = 300 GeV and Hm ¯ = 1 GeV, we get the following The largest value of the above corresponds to values for H∗ /MP and r, and respectively n = 1, the smallest possible value of θ0 , which is de2 and 3; termined by Eq. (33), when we demand that ⎧ 4/3 H∗ /MP < 10−5 . With a little −7 −2/3 , 1.4 θ0 ) √ algebra it can ⎨ (8 × 10 θ0  ⎪ H r, which gives )  0.34 be shown that (θ ∗ 0 min 3/2 , r ∼ (1 × 10−5 θ0 , 6 × 105 θ03 ) (34) (f )  O(10). Such a value of fNL may be ⎪ MP NL max ⎩ 4/3 8/3 (4 × 10−5 θ0 , 2 × 108 θ0 ) observable in the near future. In the above we assumed a reheating temperwhich is valid for the typical case of θ0 < ∼ 1, i.e. ature Treh ∼ 109 GeV, which saturates the conr< 1 for n = 1 and r

1 for n = 2, 3. ∼ straint of gravitino overproduction. This conThe value of r has to be large enough to straint is important, in the case when n = 1, avoid excessive non-Gaussianity in the density because the entropy production by the curvaton perturbation spectrum. The constraint from the decay is not enough to dilute the gravitinos. HowWMAP observations reads: r > 9 × 10−3 [24]. ever, a lower Treh corresponds to a smaller Hm ¯, Enforcing this constraint in Eq. (32) we obtain which results in a smaller r, according to Eq. (32). the bound This, in turn, results in a larger H∗ [cf. Eq. (33)], −a2 /2 −a/4   which again violates the upper bound on H∗ . On MP Hm 0.232 ¯ .(35)the other hand, a higher T θ0 > 1/(n+1) reh corresponds to the m3/2 m3/2 (103.3 πζ) ≤ m range 1 GeV < Hm ¯ 3/2 , which, for a given n, where a ≡ (n − 1)/(n + 1). Suppose first that can increase r somewhat. If r 1 then the enwe live in a typical part of the Universe, then tropy production by the decay of the curvaton reθ0 ∼ O(1) (see appendix for a discussion on the laxes the gravitino constraint. However, as shown randomization of the curvaton field). In this case by Eq. (33), H∗ loses its sensitivity to a large r, the value of r is always large enough to avoid even though H∗ may be reduced down to 10−7 MP 5 excessive non-Gaussianity. For n = 2, 3, H∗ befor Hm ¯ ∼ m3/2 . comes barely compatible with the limit H∗ /MP < 5 Note, here, that there is also a lower bound on H , 10−5 coming from the observational bound on the ∗

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D.H. Lyth / Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35

Alternatively, we may allow the possibility that θ0 is untypically small in our region. Then the value of H∗ needed to generate the observed denn+1

sity perturbation is reduced by a factor θ0 n for n = 2, 3. Now the cases n = 2, 3 become viable, but the n = 1 is not because it violates −5 the H∗ < ∼ 10 MP bound [cf. Eq. (34)]. Moroever, for n = 2 and 3 we now have the fascinating possibility that the curvature perturbation has the observed magnitude only because we live in a special part of the Universe. (To be pre−5 cise, the bound H∗ < ∼ 10 MP makes this state of affairs inevitable for n = 2, 3) This is actually a quite common outcome of curvaton models [9], and provides a good example of how a theoretical model can make anthropic considerations mandatory (in absence of a model to fix θ dynamically to a small value). In this model, Peccei-Quinn symmetry is broken during inflation and is not restored. No axionic strings are formed, and axions are produced by the oscillation of the homogeneous axion field. Their density is [16] 2 1.2   N θa fP Q D, (37) Ωa = (1 to 10) π 1012 GeV

where N is the multiplicity of the axion vacuum and D is the dilution coming from entropy production after axion creation. (In our model, D = 1 for the most interesting case n = 1.) The uncertainties are big, but it is clear that the axions can be the dark matter (Ωa = 0.25) for θa roughly of order 1. The perturbation δθa will generate a matter isocurvature perturbation Sm , which should be < 0.1ζ to be compatible with observation. It is ∼ given by Sm

=

1 δρa Ωa ρa 3

(38)

as was derived in [26]. In our case it can be shown that, because v during inflation is different that in the vacuum, this bound reads H∗ > (v∗ /fP Q )4/5 10−12 MP [27]. Using, Eqs. (16) and (25) this bound can be recast 4

as H∗ > [10−15(n+1) (MP /m3/2 )] 5n+1 MP . The tightest case corresponds to n = 1, which gives H∗ > 10−9 MP . Clearly, our results in Eq. (34) satisfy well this bound for θ0 ∼ O(1).

=

2 δθa . Ωa θa 3

(39)

But δθa ∼ δθ0 because both are generated from the vacuum fluctuation. Therefore, in view of Eq. (30), we find Sm ζ

1.2 fP Q ∼ (1 to 10) 1012 GeV    2 4 + 3r N D, θa θ0 × 3r π 

(40)

which can be < ∼ 0.1 as required by present observation. On the other hand it could be observable in the future. 6. Discussion and Conclusions Within the curvaton paradigm, we have proposed a new candidate for the field which causes the curvature perturbation. We believe that it is one of the most attractive candidates yet proposed for that field. Being a field that is part of the flaton realization of PQ symmetry-breaking, and it can easily be kept light during and after inflation. In a typical part of the universe, the observed magnitude ∼ 10−5 of the curvature perturbation comes from a modest and quite reasonable hierarchy between the inflationary Hubble parameter and the vacuum expectation value of the Peccei-Quinn field. The predicted values of the axion scale and the Hubble parameters are 1010 and 1012 GeV, respectively, corresponding to n = 1. In this case, the entropy dumping due to the curvaton decay is negligible so that the conventional cosmological predictions concerning dark matter components and baryogenesis remain unchanged. The model can generate two kinds of isocurvature pertubations. One is an axion isocurvature perturbation, uncorrelated with the curvature perturbation. The other is an isocurvature perturbation in some other kind of dark matter, or in the baryonic matter, which is produced either before or at curvaton decay. As described in [28,30], such a perturbation is generic to curvaton models, and is fully correlated or anti-correlated with the curvature perturbation. The detection

D.H. Lyth / Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35

of any such isocurvature perturbation would be evidence in support of the curvaton model. Finally, we have shown that, in this case, non-Gaussianity at at an observable level (with fNL ∼ O(10)) is possible. Our curvaton candidate is a PNGB, corresponding to an angular degree of freedom associated with the QCD axion. As a consequence, it can avoid the usual mass of order H during (and after) inflation, because supersymmetry breaking affects the mass only through A terms, which can be controlled by appropriate symmetries. Our model of PQ symmetry breaking is similar to one already extensively investigated [22,31], except that we take the PQ symmetry to be spontaneously broken throughout the history of the Universe. In contrast, the investigations of [22,31] assume that the symmetry is initially unbroken, leading to thermal inflation. These possibilities correspond respectively to radial massessquared of order ∓H 2 , and the two signs should be deemed equally likely in the absence so far of a string-theoretic prediction. The unbroken paradigm has a very different cosmology from the one that we are adopting, producing in particular copious saxion- or axino-like particles which may decay into relativistic axions whose energy density is enough to affect nucleosynthesis [22]. Also, the lightest axino-like particle may be the lightest supersymmetric particle [31], providing a more natural implementation of the Cold Dark Matter scenario that was originally proposed [32] in the context of non-flaton models. None of this occurs within our paradigm. As it invokes the axion, our model, as it stands, is open to the criticism that the axion mass is implausibly small. Indeed, in the kind of models that we have considered where the axion is an angular part of a complex field, a nonrenormalizable term in the potential with dimension d generically breaks PQ symmetry and contributes to the axion mass an amount of order ∼ v (d−2)/2 in Planck units. To keep the axion mass to the required value of order 10−30 MP , terms up to d ∼ 12 must respect the PQ symmetry. (We take v ∼ fP Q = 1012 GeV for an estimate.) A widely-discussed possibility to ensure this is to impose a Zn subgroup of the PQ

33

symmetry [34].6 Because of these considerations, one may favour a solution of the CP problem in which the QCD axion is identified with a string axion [33], or one without any axion at all.7 In that case the field considered in our proposal becomes ad hoc, introduced solely to explain the curvature perturbation. Still, because of its simplicity and the ease with which it is made sufficiently light, we feel that the present curvaton model is extremely attractive in comparison with others. Let us end by mentioning those other models which have a close connection with ours. The closely related models are considered in Refs. [12,13], where as in our case the effect of the angular part of a flat direction has been considered. The difference from our case is that the flat direction is supposed to have zero VEV, so that the curvaton does not exist as a particle in the vacuum. As a consequence, the phase can only induce the curvature fluctuations in the radial field which leads to different predictions. In addition, the proposal of [12] does not invoke the mass-squared of order −H 2 , so that the radial potential is very flat with only a two-loop thermal correction breaking the symmetry. (In both cases, successful curvature perturbations can arise for n = 3, contrary to our case with n = 1.) Because of such features the success of these models depends on computations which are much more tricky than in our case, though that is of course not necessarily an argument against them. However, we note that the computation of [12], involving the very flat radial potential, works only 6 Imposing the full PQ symmetry of course works, but exact continuous global symmetries are widely regarded as incompatible with string theory and even the existence of gravity. A different possibility, which does not seem to have been mentioned before, might be to suppose that the PQ fields are actually moduli, making the origin a point of enhanced symmetry. Then the non-renormalizable terms may be suppressed by a factor ( TeV/MP )2 ∼ 10−30 . However this is small enough for all d ≥ 5 only if fP Q has an implausibly small value of order 108 GeV. In any case, this mechanism cannot be used in the flaton case, which invokes an unsuppressed coefficient for one term. 7 The string axion as a curvaton candidate is discussed in [9]. As the string axion is a PNGB one may be hopeful that it can be kept sufficiently light in the early Universe, but string phenomenology is not yet sufficiently developed that one can be sure.

34

D.H. Lyth / Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35

if inflation lasts for a limited amount of time, because it takes the radial field during inflation to be at the edge of the slow-roll regime, V  ∼ H 2 . It seems more reasonable to assume that inflation lasts long enough to allow the quantum fluctuation to randomize the radial field within the 4 smaller region V < ∼ H . (The analogous assumption for the angular field is of course the one that we made.) The other related model [11] also invokes a flaton model of PQ symmetry breaking, but now the curvaton candidate is a radial field and it is not clear how to keep it light in the early Universe. Also, the model works only if inflation is of short enough duration that the curvaton does not enter the randomization regime. A. Appendix: Randomization PNGB curvaton

of

the

In this appendix we elaborate more on the assumption of the complete randomization of the value of the curvaton during inflation. Since the curvaton is an effectively massless field during inflation, its value is perturbed by the action of quantum fluctuations, which introduce a perturbation of the order of δσ = TH per Hubble time, where TH = H∗ /2π is the Hawking temperature of de-Sitter space. Provided that the potential V (σ) is not too steep, these quantum ‘kicks’ can move the field around by random walk so that, given enough e-foldings, the phase θ0 may assume an arbitrary value (typically of order unity) by the time when the cosmological scales exit the horizon. The criterion for this to occur for a PNGB curvaton is H∗ ≥ Hc , where √ (41) Hc ≡ mσ v.

The above critical value has been obtained by considering the fact that the region of V (σ), where the field is randomized, is determined by 4 the condition V (σ) < ∼ H∗ [35]. This condition can be understood as follows: The kinetic en4 ∼ H∗4 . ergy density of the quantum ‘kicks’ is TH Hence, the quantum fluctuations can displace the field from its minimum only up to potential den4 . For a PNGB V (σ) ∼ m2σ σ 2 , which sity V ∼ TH means that the borders of the randomized region

are at σQ ∼ H∗2 /mσ . If H∗ is large enough then the randomized region includes the entire range of σ. Setting σQ → v we obtain the critical value of H∗ , shown in Eq. (41), over which the PNGB is fully randomized. Using Eqs. (25) and (27) the bound H∗ ≥ Hc translates into  2n  H∗ n+1 ∼ (3πθ0 ζ)2 , (42) CA ≤ MP

where we also used Eq. (33) with r > ∼ 1, according to Eq. (34). With θ0 ∼ O(1) we see that complete randomization of σ∗ requires CA < 10−7 . According to Eq. (24), this condition is satisfied provided −7 8 that |I| < ∼ 10 MP . If complete randomization is not realized then σ∗ may be much smaller than v during inflation. In this case, according to Eq. (31), r can be smaller than unity and the results in Eq. (34) are modified. In particular, the typical value of σ∗ is estimated as  v for H∗ ≥ Hc , (43) σ∗  θ 0 v ∼ H∗2 /mσ for H∗ < Hc where we considered that, in the non-randomized case, the typical value is σ∗ ∼ σQ . Inserting the above into Eq. (30) and after some algebra it is easy to find (for respectively H∗ ≥ Hc and H∗ < Hc )   v r Hc /H∗ × . (44) = πζ r˜ ≡ H∗ /Hc mσ 4 + 3r

It it easy to see that r˜(r) is an increasing function of r. Hence, the WMAP non-Gaussianity constraint reads: r˜(r) > r˜(9 × 10−3 ) ∼ 2 × 10−3 . For a successful PNGB curvaton, this constraint has to be satisfied at least for the maximum allowed r˜. Therefore, in view of Eq. (44), we obtain  (45) r˜max  πζ v/mσ > 2 × 10−3 .

Using Eqs. (25), (27) and (33), we get 2 √ r˜max 8 Note

πζ r˜ CA θ0

⇒

r˜max < √

πζ . CA θ0

(46)

that, in this case, there is no problem with CA ≥ A/H∗ ∼ 10−10 .

D.H. Lyth / Nuclear Physics B (Proc. Suppl.) 148 (2005) 25–35

Combining Eqs. (45) and (46) we obtain 

πζ  0.08 θ0−1 , (2 × 10−3 ) θ0

(47)

16.

which is typically satisfied by a successful curvaton.

17. 18.

CA <

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